1 2 2 2 3 2 n 2

Now this means that the induction step "works" when ever n ≥ 3. Reduce the expression by cancelling the common factors. Join / Login. )) = 2 /(( + 1 A sum is always greater than it's smallest value times the number of terms, which in this case is $\frac{2^{k+1}-2^k}{2^{k+1}}=\frac{1}{2}$ so we are done.45 ERA in 35 games, 20 of them starts.3. Prove that 1^2 + 3^2 + 5^2 +. Tap for more steps Step 1. . .Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Solution. Step 1. *Với k = 2 thì S = 1 2 + 2 2 + 3 2 + + n 2 để tính nó thì có nhiều cách The sum of the first n n even integers is 2 2 times the sum of the first n n integers, so putting this all together gives. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.. If n 1, n 2 and n 3 are the fundamental frequencies of three segments of a string of length l, Apply that to the product $$\frac{n!}{2^n}\: =\: \frac{4!}{2^4} \frac{5}2 \frac{6}2 \frac{7}2\: \cdots\:\frac{n}2$$ This is a prototypical example of a proof employing multiplicative telescopy. and RHS = 1 6 (1 + 1)(2 +1) = 1. this involves the following steps.1,3: Prove the following by using the principle of mathematical induction for all n N: 1 + 1/((1 + 2)) + 1/((1 + 2 + 3)) + . Output: 32. A naive approach is to calculate the sum is to add every power of 2 from 0 to n. ∙ prove true for n = k + 1. Step 1. Rules for expressions with fractions: Fractions - use a forward slash to divide the numerator by the denominator, i. A term of the form f(n)g(n) can usually be converted to a L'Hopital's rule form by taking the log of both sides. Show that is true for and 2. The induction hypothesis is when n = k n = k so 3k >k2 3 k > k 2. So for the induction step we have n = k + 1 n = k + 1 so 3k+1 > (k + 1)2 3 k + 1 > ( k + 1) 2 which is equal to 3 ⋅3k > k2 + 2k + 1 3 ⋅ 3 k > k 2 + 2 k + 1. This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. This update, iOS 17. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.sreerac rieht dliub dna ,egdelwonk rieht erahs ,nrael ot srepoleved rof ytinummoc enilno detsurt tsom ,tsegral eht ,wolfrevO kcatS gnidulcni seitinummoc A&Q 381 fo stsisnoc krowten egnahcxE kcatS :hcaorppa evoba eht fo noitatnemelpmi eht si woleB . Let n in 2^n be 1, or 2^1 = 2.org or mail your article to review-team@geeksforgeeks. 想像一个有圆圈构成的正三角形，. Xem lời giải. Assuming the statement is true for n = k: 12 + 22 + 32 + + k2 = k(k + 1)(2k + 1) 6; (1) we will prove that the statement must be true for n = k + 1: A Computer Science portal for geeks. HOC24. The term before in the sum will be half of 2, so we can also write the entire sum as: Find the sum of the series $$1^2-2^2+3^2-4^2+-(2n)^2$$ I tried rewriting it as $$\sum_{r=1}^{2n}-1^{n+1}(r^2)$$ but it didn't help. A new variant of the virus that causes COVID-19 is rising to prominence in the U. Please Enter any Positive Number : 7 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 140. The y-intercept of the parabola is − + 1 / 12. The associated homogeneous recurrence relation is an = 2an−1 a n = 2 a n − 1 . For math, science, nutrition, history You are trying to understand why. Related. ∙ assume the result is true for n = k. Summing integers up to n is called "triangulation". My Notebook, the Symbolab way. H. {an}n=1n=10, an = n2. An example of a negative mixed fraction: -5 1/2. (What you wrote, 1+ 2^1+ + 2^n-1= 2^n-1 is, as Ray Vickson said, clearly impossible because you have "2^n- 1" on both sides but with additional positive terms on the left.. \bold{=} + Go. Tính các giá trị của biểu thức T = a 2 + b 2 A.56 billion) and new shares issued by Harbour equating to a total shareholding in the enlarged Harbour of 54. + 2^n = 2^{n + 1} - 1 \forall n \in N\] \[\text{ Step I: For } n = 1, \] \[LHS = 1 + 2^1 = 3\] \[RHS = 2^{1 + 1} - 1 = 2 $$=n^3+n^2(n+1)+\frac{n(n+1)(2n+1)}6=\ldots$$ Share. + 2 n. DonAntonio DonAntonio. This is because you can think of the sum as the … Repeat the process until your list is empty - you now have N/2 pairs of numbers that each add to N+1. For n ≥ 0 n ≥ 0, let S(n) S ( n) denote the statement. + n 2 = n n + 1 2 n + 1 6. Steps {3}{2^n} Show More; Description. Share 7. So you will get 2^2-1 = 3. Học bài Hỏi bài Giải bài tập Đề thi ĐGNL Khóa học Tin tức Cuộc thi vui Tìm kiếm câu trả lời Tìm kiếm câu trả lời cho câu hỏi của bạn; Đóng Explanation: using the method of proof by induction. 1 Answer Solve an equation, inequality or a system. Prove the following by using the principle of mathematical induction for all n ∈ N. He has been teaching from the past 13 years. Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Math notebooks have been around for hundreds of years.1,3: Prove the following by using the principle of mathematical induction for all n N: 1 + 1/((1 + 2)) + 1/((1 + 2 + 3)) + .. So, the Geometric mean G... The method of regularization using a cutoff function can "smooth" the series to arrive at − + 1 / 12. 第一行1个圈，圈内的数字为1.5% (BASF share: 39. For math, science, nutrition, history Mình xin nói một cách tổng quát về bài toán tính tổng S = 1 k + 2 k + 3 k + + n k như sau: Đầu tiên, với k = 1 thì S = 1 + 2 + 3 + + n cái này thì ai cũng biết công thức và cách chứng minh rồi : S = n ( n + 1) 2. Suppose we take 2^n in the sum. View Solution. Aug 23, 2011 at 10:01 2 (n + 1)3 −n3 = 3n2 + 3n + 1 - so it is clear that the n2 terms can be added (with some lower-order terms attached) by adding the differences of cubes, giving … Example 1 For all n ≥ 1, prove that 12 + 22 + 32 + 42 +…+ n2 = (n(n+1)(2n+1))/6 Let P(n) : 12 + 22 + 32 + 42 + …. Examples: Input : n = 4 Output : 84 Explanation : sum = 1 2 + 3 2 + 5 2 + 7 2 = 1 + 9 + 25 + 49 = 84 Input : n = 10 Output : 1330 Explanation : sum = 1 2 + 3 2 + 5 2 + 7 2 + 9 2 + 11 2 + 13 2 + 15 2 + 17 2 + 19 2 = 1 + 9 + 24 + 49 + . Similarly, if an =∑n k=1 k a n = ∑ k = 1 n k, then an = n(n + 1)/2 a n = n ( n + 1) / 2 is given by a quadratic polynomial.02. Lớp học.1009. 1 1 + 2 2 + 3 3 = 1 + 4 + 27 = 32. = n(n)(n+1) 2 − n(n+1)(2n+1) 6 + n(n+1) 2.Aug 23, 2011 at 10:01 2 (n + 1)3 −n3 = 3n2 + 3n + 1 - so it is clear that the n2 terms can be added (with some lower-order terms attached) by adding the differences of cubes, giving a leading term in n3. Summing integers up to n is called "triangulation". \sum_ {k=1}^n (2k-1) = 2\sum_ {k=1}^n k So when you get to $1,2,2,3$ this can only go to $1,2,2,3,2,3,3,4$. Limits. 以此类推. Hence, the sum of all integers from 1 to an even N is (N+1)*N/2.4) + 7/(3. Cite. \frac {2n (2n+1)}2 - 2\left ( \frac {n (n+1)}2 \right) = n (2n+1)-n (n+1) = n^2. We can expand this inequality $(n-1)^2>2$ as follows: \begin{align*} n^2-2n+1>&\,2\\ n^2-2n-1>&\,0\\ 2n^2-2n-1>&\,n^2\\ 2n^2>&\,n^2+2n+1=(n+1)^2, \end{align*} which is the second inequality claimed in $(\spadesuit)$. We use power function to compute power.. Học bài Hỏi bài Giải bài tập Đề thi ĐGNL Khóa học Tin tức Cuộc thi vui Tìm kiếm câu trả lời Tìm kiếm câu trả lời cho câu hỏi của bạn; Đóng Click here:point_up_2:to get an answer to your question :writing_hand:displaystyle limn rightarrow inftydfrac 122232n2n3 is equal to Chứng minh rằng: A = 3/(1. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.+ n^2 = n(n + 1)(2n + 1)/6 for n greaterthanorequalto 1. 12 + 22 + 32 + + n2 = n(n+ 1)(2n+ 1) 6 Proof: For n = 1, the statement reduces to 12 = 1 2 3 6 and is obviously true. Use app Login. Take three of the rows, and remove them. New numerator is 6 + 2 = 8. But in this Python program , we are defining a Functions to place logic.1. Sum of the series 1 1 2 2 3 3 n n using recursion in C - In this problem, we are given a number n which defines the nth terms of the series 1^1 + 2^2 + 3^3 + … + n^n.0 This Python Sum of Series 1²+2²+3²+….Let’s take an example to understand the problem,Input n = 4Output30Explanation −sum = (1^1) + (2^2) + (3^3) + … Not a general method, but I came up with this formula by thinking geometrically. Hence, the n -th term of the series is S n = ∑ n = 1 n 2 n - 2 n + 1.4. 1: 2: 3: 4: 5: 6: 7: 8: 9: 0.e. A naive approach is to calculate the sum is to add every power of 2 from 0 to n. Q5. .. But it is easier to use this Rule: x n = n (n+1)/2. Study Materials. He has been teaching from the past 13 years. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Base step (n = 0 n = 0 ): S(0) S ( 0) says that 20 = 21 − 1 2 0 = 2 1 − 1, which is true. Our task is to create a program that will find the sum of the series. Step 2.geeksforgeeks. Click here:point_up_2:to get an answer to your question :writing_hand:the value of 1122 33 nn is If n 1, n 2 and n 3 are the fundamental frequencies of three segments into which a string is divided, then the fundamental frequency n of the original string is given by. Multiply the exponents in .3) + 5/(2. Input: n = 2 Output: -3 Explanation: sum = 1 2 - 2 2 = 1 - 4 = -3 Input: n = 3 Output: 6 Explanation: sum = 1 2 - 2 2 + 3 2 = 1 - 4 + 9 = 6 Naive Approach: This method involves simply running a loop of i from 1 to n and if i is odd then simply add its square to the result it i is even then simply subtract square of it to the result.. S: 1 3 = 1 R. Was this answer helpful? Asymptotic behavior of the smoothing. M = 2 n 2 [ ∵ Since the sum of n natural numbers is n Imagine a big square of dots. Input: n = 3. Given sequence, 2 1 + 2 2 + 2 3 +.+n2 = n(n+1)(2n+1) 6 P (1): 12 = 1(1+1)(2(1)+1) 6 1 = 6 6=1 ∴ LH S =RH S Assume P (k) is true P (k): 12 +22 +32+. Guides. - Steve Jessop. #sum_(n=1)^(N) (-1)^(n+1) n^2# 3. Verified by Toppr. DERIVATION. So for your case. So for your case. H. Bernard's answer highlights the key algebraic step, but I thought I might mention something that I have found useful when dealing with induction problems: whenever you have an induction problem like this that involves a sum, rewrite the sum using -notation. this involves the following steps. Pair it up with our Nike Swoosh fleece pants for a uniform look, heavy on the Swoosh. Use iteration to solve the recurrence relation with. December 18, 2023 12:17 PM EST. . 4. 2^ (2n) can be expressed as (2^n) (2^n), and 2^n isn't a constant. . View Solution. Fixes include resolving multiple crashes, freezes, removal of invisible walls, stability improvements, issues with the Na'vi senses feature, and balancing..4. Question: Prove that 1^2 + 2^2 + 3^2 +.We can find the sum of squares of the first n natural numbers using the formula, SUM = 1 2 + 2 2 + 3 2 + + n 2 = [n(n+1)(2n+1)] / 6. n = 1 → LH S = 12 = 1. Use the formula of the sum of the first n natural numbers. ⇒ S 2 = 2 2 - 2 3 ⇒ S 3 = 2 3 - 2 4 ⋮ ∴ S n = 2 n - 2 n + 1.Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler-Maclaurin formula.. Tap for more steps Step 1. What is the value of the sum: #(1^2)+(1^2+2^2)+(1^2+2^2+3^2)+.upto n terms will be. Prove that. . + 1/((1 + 2 + 3 + .Else, calculate the sum of squares recursively by adding n*n with the sum_of_squares of n-1. $1 + 3 + 3^2 + + 3^{n-1} = \dfrac{3^n - 1}2$ I am stuck at $\dfrac{3^k - 1}2 + 3^k$ and I'm not sure if I am right or not. 3. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. When describing sequences, the following notation is standard: \ {a_n\}_ {n=1}^ {n=10}, \quad a_n = n^2. 想像一个有圆圈构成的正三角形，. It is the smallest and only even prime number. M = 2 n ( n + 1) 2 1 n + 1 ⇒ G. Below is the implementation of the above approach: Với mọi số nguyên dương n ≥ 2, ta có: 1 − 1 4 1 − 1 9 1 − 1 n 2 = an + 2 bn, trong đó a, b là các số nguyên. Thus, in general, the sum of the series can be Let us first recall the meaning of natural numbers. #1 * 1! + 2 * 2! + 3 * 3! = 1+4+18 = 23# Note that we should expect a sum that involves a factorial somewhere and #23 = 24-1 = 4!-1# . . The factor 1/3 attached to the n3 term is also obvious from this observation. Share. Given an integer N, the task is to find the sum of series 2 0 + 2 1 + 2 2 + 2 3 + ….1 ot lauqe ro retaerg sregetni lla rof $2^n > n^3$ ces_elpirt . Share 7. Simplify (2^(n+1))/((2^n)^(n-1)) Step 1.It may be written in mathematics as or /. Divide by . a) Multiply the whole number 2 by the denominator 3. Limits.nigoL ppa esU . It makes everything more concise and easier to manipulate: ∑i=1k+1 i ⋅ i! =∑i Given a series 1 2 + 3 2 + 5 2 + 7 2 + .

 Hence, the sum of the series, when the number of terms is odd, is n 2 + n 2

., < > ≤: ≥ ^ √: ⬅: : F _ ÷ | (* / ⌫ A: ↻: x: y = +-G Mathematics Proof by mathematical induction Question Prove by mathematical induction, 12 +22 +32+. Arithmetic.For any value N-Given 1^2, (1^2+2^2), (1^2+2^2+3^2),…. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. JavaScript has been disabled on your browserenable JS. He moved from the rotation to the bullpen in August and made three relief appearances in Favorite. + 361 = 1330 What is the value of the sum: #(1^2)+(1^2+2^2)+(1^2+2^2+3^2)+. A power of two is a number of the form 2 n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent. View Solution. Explanation −.

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Then using that value, the compiler will find the sum of series 1 2 + 2 2 + 3 2 + … + n 2 using the above formula. + (2n - 1) 2, find sum of the series.. lndn = ln((1 + 2 n)n) = n ln(1 + 2 n) = ln(1 + 2 n) 1 n. Question 1 Important Deleted for CBSE Board 2024 Exams Question 2 Deleted for CBSE Board 2024 Exams Question 3 Important Deleted for CBSE Board 2024 Exams Question 4 The sum of the series 1+1+2+1+2+3+. A sequence is an ordered list of numbers.1., 1 2/3 . To see how this works, let's go through the same example we used for telescoping, but this time use iteration. 第二行2个圈，圈内的数字都为2，. For loop is used to compute the sum of series. Given sequence, 2 1 + 2 2 + 2 3 +. Mathematics. \frac {2n (2n+1)}2 - 2\left ( \frac {n (n+1)}2 \right) = n (2n+1)-n (n+1) = n^2. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Simultaneous equation. It has rows and columns. Whole number 2 equally 2 3. Style: DX0566-657. S ( n): ∑ i = 1 n 2 i = 2 n + 1 − 1. Examples: Input : n = 3 Output : 1. It is clear that the given geometric progression has n + 1 terms.4142) is a positive real number that, when multiplied by itself, equals the number 2.+ (2n + 1)^2 = (n + 1)(2n + 1)(2n Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their … This question already has answers here : Proving 13 +23 + ⋯ +n3 =(n(n+1) 2)2 1 3 + 2 3 + ⋯ + n 3 = ( n ( n + 1) 2) 2 using induction (16 answers) Closed 9 years ago. Arithmetic. From here you can probably show that. However to start the induction you need something greater than three. + 2 n forms a GP with first term a = 2 and common ratio r = 2 Since, sum of n terms of GP = a (r n 1: 2: 3-\pi: e: x^{\square} 0. P = 5 \[Let p\left( n \right): 1 + 2 + 2^2 + . 3n >n2 3 n > n 2.+ 2^n. However, constant factors are the only thing you can pull out. Please let me know how to improve the proof and if I got it really wrong what the right answer is. The equation calculator allows you to take a simple or complex equation and solve by best method possible. Simultaneous equation. In exchange, at closing, the shareholders of Wintershall Dea - BASF (72. limn→∞ lndn = 2.#upto n terms? Precalculus Series Summation Notation. Viewed 14k times 4 $\begingroup$ I am wondering if the third to last equation is correct, where i factored out the $(-1)^k$.Check if n is 1, return 1. Prove that 1 2 +2 2 +3 2 +4 2 +··· + n 2 = n (n+1) (2n+1) 6 Here is source code of the C Program to Find the Sum of Series 1/1! + 2/2! + 3/3! + ……1/N!. Tap for more steps Step 2. The characteristic equation is r − 2 = 0 r − 2 = 0 . You have been given a series 1 + 1/2^2 + 1/3^3 + …. sum = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 = 84. Prove that 1 2 +2 2 +3 2 +4 2 +··· + n 2 = n (n+1) (2n+1) 6 for every positive integer n. This is because you can think of the sum as the number of dots in a stack where n dots are on the bottom, n-1 are in the next row, n-2 are in the next row, and so on.Call sum_of_squares function with N as input and store the result in sum_of_squares variable. 1.28704 Explanation : 1 + 1/2^2 + 1/3^3 Input : n = 5 Output : 1. Our task is to create a program that will find the sum of the series. M = 1 · 2 · 2 2 ·. Answer. See your article appearing on the GeeksforGeeks main page and help other Geeks.+n² program is the same as above.1009. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions. and RHS = … Prove that $1^2-2^2+3^2-…+(-1)^{n-1} n^2$=$(-1)^{n-1}\frac{ n(n+1)}{2}$ whenever n is a positive integer using mathematical induction. (N-1) + 1 + (N-2) + 2 + The way the items are ordered now you can see that each of those pairs is equal to N (N-1+1 is N, N-2+2 is N). Share Cite answered Oct 18, 2014 at 15:07 Brad 100 1 9 where did the (−1)k ( − 1) k go between lines 1 and 2 Sep 15, 2022 at 11:33 Add a comment Explanation: using the method of proof by induction. n = 5. Then looking at the previous values we have #5 = 6-1 = 3!-1# and #1 = 2-1 = 2!-1# Answers archiveAnswers Question 229820: Answer by ( Show Source ): You can put this solution on YOUR website! prove 1. Tap for more steps 2n+1−n2+n 2 n + 1 - n 2 + n. Initialize the value of 'i Approach: The sequence is formed by using the following pattern. 1 1 + 2 2 + 3 3 = 1 + 4 + 27 = 32. Solve your math problems using our free math solver with step-by-step solutions. Find nth Term of the Series 1 2 2 4 4 4 4 8 8 8 8 8 8 8 8 Find the Nth term of the Zumkeller Numbers; Find Nth term of the series where each term differs by 6 and 2 alternately; Practical Numbers; Find value of (1^n + 2^n + 3^n + 4^n ) mod 5; Zygodrome Number; Gapful Numbers; Program to find sum of series 1 + 2 + 2 + 3 + 3 + 3 + . Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. The sum of the first n n even integers is 2 2 times the sum of the first n n integers, so putting this all together gives. H. Sum of series = 1^2 + 2^2 + ….org. It is the natural number following 1 and preceding 3. I am using induction and I understand that when n = 1 n = 1 it is true. It's a couple steps more to show that this also works for odd N, and that you get the formula you asked about if you replace N with N-1. Calculate the sum. Sum of the series 1 1 2 2 3 3 n n using recursion in C - In this problem, we are given a number n which defines the nth terms of the series 1^1 + 2^2 + 3^3 + … + n^n. ∙ prove true for some value, say n = 1. The sum of a geometric series is given by the formula: S = a (1 - r^n)/ (1 - r) where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms. Solve your math problems using our free math solver with step-by-step solutions. 2. Of course, one reason for creating the digamma function is to make formulae like this true.:#)n(^)1-(# a edulcni ot mus eht etirw nac ew ,gnitanretla si seires eht ecniS #2^N + . )) = 2 /(( + 1.6%). 2n+1 (2n)n−1 2 n + 1 ( 2 n) n - 1.. 1 2 + 3 2 + 5 2 + ⋯ + (2 n − 1) 2 = n (2 n − 1) (2 n + 1) 3 View Solution Q 4 1. + 361 = 1330 1 1 + 2 2 = 1 + 4 = 5. Induction step (S(k) → S(k + 1) S ( k) → S ( k + 1) ): Fix some k ≥ 0 k ≥ 0 and suppose that.Tech from Indian Institute of Technology, Kanpur. . 另外一个很好玩的做法. View Solution. Assume is true for some positive integer , then show the relationship is true for , namely that: First note that: which can be written: .S. Of course, you meant 2^(n-1) on the left and (2^n)- 1 on the right. Prove that 1^2 + 3^2 + 5^2 +. + n^2= n (n + 1) (2n + 1) / 6.+ n2 = (𝑛(𝑛 + 1)(2𝑛 + 1))/6 Proving 1 Answer Sorted by: 1 Your proof is completely correct. 以此类推.+ (2n + 1)^2 = (n + 1)(2n + 1)(2n Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.459, and then the factorial becomes much greater. Now reorder the items so, that after the first comes the last, then the second, then the second to last, i.2. 3n >n2 3 n > n 2.1为字数的内圈，圈个1行一第 . But And John By Jamie Ducharme.Câu hỏi trong đề: Giải toán 11: Trả lời: Giải bởi Vietjack + Với n = 1 : ⇒ (3) đúng với n = 1 + Giả sử đẳng thức (3) đúng với n = k nghĩa là : Cần chứng minh (3) đúng khi n = k + 1, tức là: Thật vậy: 3 Answers Sorted by: Reset to default 2 $\begingroup$ $2^n + 2^n = 2^n(1+1) = 2^n(2) = 2^{n+1}$ If you realise that there are $2$ of $2^n$, then we have $$2^1\times2^n$$ If we are multiplying $2$ by itself n times and then multiplying the result by another $2$, we get $2$ multiplied by itself n+1 times, which is $$2^{n+1}$$ Share. The agreed enterprise value for the ChatGPT and Microsoft Copilot are both artificial intelligence (AI) technologies that were developed with the intent of helping you accomplish tasks and activities faster and more efficiently. NCERT Solutions for Class 10 Science Chapter 1; NCERT Solutions for Class 10 Science Chapter 2; NCERT Solutions for Class 10 Science Chapter 3 Hint $ $ First trivially inductively prove the Fundamental Theorem of Difference Calculus $$\rm\ F(n) = \sum_{k\, =\, 1}^n f(k)\, \iff\, F(n) - F(n\!-\!1)\, =\, f(n Sum: 2. You can also see that the midpoint of r and s corresponds to … The sum of the first n n even integers is 2 2 times the sum of the first n n integers, so putting this all together gives. ∙ assume the result is true for n = … Question: Prove that 1^2 + 2^2 + 3^2 +. A series is the sum of the terms of a sequence. One can write $$1+\frac12+\frac13+\cdots+\frac1n=\gamma+\psi (n+1)$$ where $\gamma$ is Euler's constant and $\psi$ is the digamma function. GTU PPS Practical - 25 Write a program to evaluate the series 1^2+2^2+3^2+……+n^2 #include int main() { int n, i, sum = 0; printf("n Enter Value of n : "); A geometric progression 1, 2, 2 2,. 1. $\begingroup$. Reduce the expression by cancelling the common factors. 211k 17 17 gold badges 135 135 silver badges 287 287 bronze badges $\endgroup$ 4 $\begingroup$ Wow thanks for this detailed solution! 1/2+2/3 Final result : 7 — = 1. The C program is successfully compiled and run on a Linux system. Plus there's one more dot. . Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ? Stack Exchange Network.1010) 5/4 Nâng cao phát triển và bồi dưỡng Toán lớp 6 qua 22 chuyên đề; Đề thi HSG môn Toán lớp 7 cấp Quận/ Huyện (có giải chi tiết) Đề thi học kì 2 môn Toán lớp 6 quận Gò Vấp – TP HCM (N-1) + (N-2) ++ 2 + 1 is a sum of N-1 items. - Giả sử đẳng thức đúng với n = k ≥ 1, nghĩa là: Ta phải chứng minh rằng đẳng thức cũng đúng với n = k + 1, tức là: Thật vậy, từ giả thiết quy nạp ta có: Vậy đẳng thức đúng với mọi n ∈ N* Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals.It is an algebraic number, and therefore not a transcendental number.2 iPhone update appeared on Thursday, November 30, 2023. + 2 n forms a GP with first term a = 2 and common ratio r = 2 Since, sum of n terms of GP = a (r n Prove using the technique of "Mathematical Induction" . Prove the following by using the principle of mathematical induction for all n ∈ N 1 2 + 1 4 + 1 8 + ⋯ + 1 2 n = 1 − 1 2 n.) - Khi n = 1, VT = 1; ⇒ VT = VP , do đó đẳng thức đúng với n = 1. What is the value of $21^2 + 22^2 + \cdots + 40^2$? Using induction, how can I solve this problem? Stack Exchange Network. If you like GeeksforGeeks and would like to contribute, you can also write an article using write. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. Auxiliary Space: O(1) for constant space for variables 6 Answers. So, the answer to your questions are yes and no.70833. Add n n and n n. You write down problems Add a comment. Even more succinctly, the sum can be written as.Let's take an example to understand the problem,Input n = 4Output30Explanation −sum = (1^1) + (2^2) + (3^3) + (4^4 Repeat the process until your list is empty - you now have N/2 pairs of numbers that each add to N+1. S N = N * (N+1) 2 * (N+2) / 12. 22n(2n+1) −2( 2n(n+1)) = n(2n+1)− n(n+ 1) = n2. limn→∞dn =e2.. Solve your math problems using our free math solver with step-by-step solutions. There is the same number of rows as columns. Click here:point_up_2:to get an answer to your question :writing_hand:displaystyle limn rightarrow inftydfrac 122232n2n3 is equal to Chứng minh rằng: A = 3/(1. If all the terms were adding, the sum would be: #sum_(n=1)^(N) n^2 = 1^2 + 2^2 + . 第n行n个圈，圈内的数字都为n，. Standard XII. S: (1)2 = 1 R.3. M is as follows: G. For math, science, nutrition, history Mình xin nói một cách tổng quát về bài toán tính tổng S = 1 k + 2 k + 3 k + + n k như sau: Đầu tiên, với k = 1 thì S = 1 + 2 + 3 + + n cái này thì ai cũng biết công thức và cách chứng minh rồi : S = n ( n + 1) 2. Also, looked at re-arranging as $$1^2+3^2+5^2+7^2++(2n-1)^2$$ and $$-2^2-4-6^2-8^2--(2n)^2$$ Still couldn't get to the given answer of $-n(2n+1)$ Solve your math problems using our free math solver with step-by-step solutions. .5) + … + 2017/(1008. So for the induction step we have n = k + 1 n = k + 1 so 3k+1 > (k + 1)2 3 k + 1 > ( k + 1) 2 which is equal to 3 ⋅3k > k2 + 2k + 1 3 ⋅ 3 k > k 2 + 2 k + 1.e. Sum of the Series 1/(12) + 1/(23) + 1/(34) + 1/(45) + . Visualization of powers of two from 1 to 1024 (2 0 to 2 10). ∑n1 i2 = n(n + 1)(2n + 1) 6, (1) (1) ∑ 1 n i 2 = n ( n + 1) ( 2 n + 1) 6, which is in fact very well known--just google something like "sum of first n squares"; you'll get about a gazillion hits. Hint only: For n ≥ 3 you have n2 > 2n + 1 (this should not be hard to see) so if n2 < 2n then consider 2n + 1 = 2 ⋅ 2n > 2n2 > n2 + 2n + 1 = (n + 1)2. 1 2 + 3 2 + 5 2 + Sum of the series 2^0 + 2^1 + 2^2 +…. Improve this answer. this involves the following steps. \sum_ {k=1}^n (2k-1) = 2\sum_ {k=1}^n k Step 1: Enter the Equation you want to solve into the editor. Guides. step-by-step. fraction and use a forward slash to input fractions i. Example: 2x-1=y,2y+3=x. 7.1. Standard XII. + 1/((1 + 2 + 3 + . (N-1) + 1 + (N-2) + 2 + The way the items are ordered now you can see that each of those pairs is equal to N (N-1+1 is N, N-2+2 is N). The printf statement will ask the user to enter any integer value. .4. You can probably arrange things so that you always access your stored values sequentially, not sure. While they may seem similar, there are significant differences between the two.. Ex 4. This is what I've been able to do: Base case: n = 1 n = 1 L. O (2^ (n+1)) is the same as O (2 * 2^n), and you can always pull out constant factors, so it is the same as O (2^n). Oct 1, 2009 at 11:59.459 x ≈ 3. H. Integration. Differentiation.e. Inductive Step to prove is: 2 n + 1 = 2 n + 2 − 1 Our hypothesis is: 2 n = 2 + 1 1 Here is where I'm getting off track. Verified by Toppr. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures .. Với k = 2 thì S = 1 2 + 2 2 + 3 2 + + n 2 để tính nó thì có nhiều cách So when you get to $1,2,2,3$ this can only go to $1,2,2,3,2,3,3,4$.. c) Write a previous answer (new numerator 8) over the denominator 3. This is what … Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}-3 = -\frac{3}{2}.2. + (2*n – 1) 2, find sum of the series. Modified 3 years, 5 months ago. S(n): ∑i=1n 2i =2n+1 − 1. 3. In the arithmetic sequence example, we simplified by multiplying by the number of times we add it to when we get to to get from to. C++ One and one half is three halfs. Step 3: Calculate the sum of the first n natural number. Cite. A basic approach to solve this problem is by directly applying the formula for the sum You'll get a detailed solution from a subject matter expert that helps you learn core concepts.1010) 5/4 Nâng cao phát triển và bồi dưỡng Toán lớp 6 qua 22 chuyên đề; Đề thi HSG môn Toán lớp 7 cấp Quận/ Huyện (có giải chi tiết) Đề thi học kì 2 môn Toán lớp 6 quận Gò Vấp - TP HCM (N-1) + (N-2) ++ 2 + 1 is a sum of N-1 items.

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+n^2. It makes everything more concise and easier to manipulate: ∑i=1k+1 i ⋅ i! =∑i Given a series 1 2 + 3 2 + 5 2 + 7 2 + . M = 2 1 + 2 + 3 + + n 1 n + 1 ⇒ G. An efficient approach is to find the 2^ (n+1) and subtract 1 from it since we know that 2^n can be written as: Feeling lost Tính tổng:S = 1^2+2^2+3^2+. 另外一个很好玩的做法. Output −. as winter illness season approaches its peak: JN. ∙ prove true for some value, say n = 1. An efficient approach is to find the 2^ (n+1) and subtract 1 from it since we know that 2^n can be written as: Feeling lost Tính tổng:S = 1^2+2^2+3^2+. NCERT Solutions For Class 12. Time Complexity: O(n) Auxiliary Space: O(1), since no extra space has been taken., for five-hundredths, enter 5/100. Prove the following by using the principle of mathematical induction for all n ∈ N. b) Add the answer from the previous step 6 to the numerator 2.+n2 = n(n+1)(2n+1) 6 Solution Verified by Toppr P (n): 12 +22 +32+. I am using induction and I understand that when n = 1 n = 1 it is true. If you use mixed numbers, leave a space between the whole and fraction parts. Open in App. Since there are N-1 items, there are (N-1)/2 such Davneet Singh has done his B. Apply the distributive Linear equation. Base Case: let n = 0 Then, 2 0 + 1 − 1 = 1 Which is true. Visit Stack Exchange 3 (1^2+2^2++n^2)=n^3+n^2+n (n+1)/2= (n/2) (2n^2+2n+n+1) = (n/2) (n+1) (2n+1) 1^2+2^2+3^2++n^2=n (n+1) (2n+1)/6.Prove that 1^2+2^2+3^2+4^2+…n^2=(n(n+1)(2n+1))/6 for every positive integer n.. 2 ( two) is a number, numeral and digit. Prove the following by using the principle of mathematical induction for all n ∈ N. + n^2 using 'number' integer variable.e. simplify \frac{(n+1)^{2}}{(n+2)^{2}} en., 2 n is given. The brute force approach: We have. The square root of 2 (approximately 1.1. If we consider n consecutive natural numbers, then finding the sum of the squares of these numbers is represented as Σ i = 1 n i 2. Example 1 For all n ≥ 1, prove that 12 + 22 + 32 + 42 +…+ n2 = (n(n+1)(2n+1))/6 Let P(n) : 12 + 22 + 32 + 42 + ….2. 22n+1−n2 2 2 n + 1 - n 2. If n ∈ N, then 1·2+2·3+3·4+4·5+··· + n (n+1) = n (n+1) (n+2) 3 . Login. Mathematics.7%) and LetterOne (27. Even more succinctly, the sum can be written as. Q5. ⇒result is true for n = 1. In a context where only integers are considered, n is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. Find S 1, S 2, S 3, ⋯, S n to calculate the sum of the series.,till N terms. Solve. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Then, since ln is continuous, limn→∞ lndn = ln limn→∞dn = 2, and you can solve to get. `Made with soft fleece in a roomy fit for casual comfort, this Nike Swoosh 1/2-zip hoodie brings the bold Nike vibes to any outfit`. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals.2. Since our characteristic root is r = 2 r = 2, we know by Theorem 3 that an =αn2 a n = α 2 n Note that F(n) = 2n2 F ( n) = 2 n 2 so we know by Theorem 6 that s = 1 s = 1 and 1 1 is not a root, the Click here:point_up_2:to get an answer to your question :writing_hand:the value of 12 22 32 n2 is 3.+n^2.#upto n terms? Precalculus Series Summation Notation. Click here:point_up_2:to get an answer to your question :writing_hand:212223 2n. Example: the 5th Triangular Number is x 5 = 5 (5+1)/2 = 15, Click here:point_up_2:to get an answer to your question :writing_hand:212223 2n. Prove the following by using the principle of mathematical induction for all n ∈ N. · 2 n 1 n + 1 ⇒ G. 第n行n个圈，圈内的数字都为n，. 1 Answer Similarly, if an =∑n k=1 k a n = ∑ k = 1 n k, then an = n(n + 1)/2 a n = n ( n + 1) / 2 is given by a quadratic polynomial. 5. Within the main() function, We declared 2 integer variables Number and Sum. Solve. Solution. To compute the sum of series, the following formula is used. 84.1, yet The unexpected iOS 17.. Step 2: Click the blue arrow to submit and see the result! The equation solver allows you to enter your problem and solve the equation to see the result. Open in App.Define a function sum_of_squares (n) which takes an integer n as input.13 +23 +33+⋯+n3 =( n(n+1) 2)2. The first part of this description, \ {a_n\}_ {n=1}^ {n=10} {an}n=1n=10, could be expanded as a list like this: a_1, a Our task is to find the sum of series 1^2 + 3^2 + 5^2 + + (2n - 1)^2 for the given value of n.2. If you already know a^m and a^a for all a less than m, then when you come to calculate (m+2)^ (m+2) then it's just 2^ (m+2) = 2^m2^2. . Base Case: let n = 0 Then, 2 0 + 1 − 1 … Prove the following by using the principle of mathematical induction for all n ∈ N 1 2 + 1 4 + 1 8 + ⋯ + 1 2 n = 1 − 1 2 n.. = n(n+1) 6 (3n−(2n+1)+3) [taking n(n+1) 6 as common from the 3 terms] = n(n+1)(n+2) 6.+ n2 = (𝑛(𝑛 + 1)(2𝑛 + … Explanation: using the method of proof by induction. Related Symbolab blog posts. Below is the implementation of the above approach: Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.91667. Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}-3 = -\frac{3}{2}.. HOC24. Sum of all natural numbers in range L to R Sum of numbers from 1 to N which are in Lucas Sequence In this C program, the user asked to enter any positive integer.3%) - will receive total cash consideration of $2. + 2 n. Sum, S =∑n r=1 r(n−(r−1)) ⇒ S= ∑n r=1rn−∑n r=1r2 +∑n r=1 r. In the statement of the problem we see $1,2,2,3$ but we don't see the next $4$ numbers, which are the solution. Notice how much simpler the proof becomes after transforming into a form where the induction is obvious, namely: $\:$ a product is $>1$ if all factors In this C Program, we are reading the limit to compute summation from the series 1^2 + 2^2 + …. Step 2: … 33 How do I prove this by induction? Prove that for every natural number n, 2 0 + 2 1 + + 2 n = 2 n + 1 − 1 Here is my attempt. Keep reading to see how these tools are powered by AI and what role they Pérez went 10-4 for the Rangers last season, going 10-4 with a 4. S: ( 1) 2 = 1 Therefore it's true for n = 1 n = 1. Solve problems from Pre Algebra to Calculus step-by-step . Step 2. = - 1 n - 1 n - 1 + 1 2 + n 2 = - n - 1 n 2 + n 2 = - n 2 - n 2 + n 2 = - n 2 + n + 2 n 2 2 = n 2 + n 2. Alternatively, plot x! −2x x! − 2 x to see a demonstration of the difference. . It’s a couple steps more to show that this also works for odd N, and that you get the formula you asked about if you replace N with N-1. ∙ prove true for some value, say n = 1. Then (m+3)^ (m+3) = 3^m3^3 and so on. Less than two weeks later, here's the next release, warning all users to update now. 第二行2个圈，圈内的数字都为2，.+ 2^n. Hence, the sum of all integers from 1 to an even N is (N+1)*N/2. Approach: Starting from n, start adding all the terms of the series one by one with the value of n getting decremented by 1 in each recursive call until the value of n = 1 for which return 1 as 11 = 1.+k2 = k(k+1)(2k+1) 6 P (k+1) is given by, P (k+1): Solution Verified by Toppr Let Sn =12 +22 +⋯ +n2 Consider the identity k3 −(k−1)3 =3k2 −3k+1 Putting k =1,2,. Since there are N-1 items, there are (N-1)/2 such Davneet Singh has done his B. In the statement of the problem we see $1,2,2,3$ but we don't see the next $4$ numbers, which are the solution.noitaitnereffiD .29126 Explanation : 1 + 1/2^2 + 1/3^3 + 1/4^4 + 1/5^5. i. + n The series 1/a + 2/a^2 + 3/a^3 + … + n/a^n is a geometric series with first term 1/a and common ratio 1/a. Integration. \frac {2n (2n+1)}2 - 2\left ( \frac {n (n+1)}2 \right) = n (2n+1)-n … Step 1: Enter the Equation you want to solve into the editor.16667 6 Step by step solution : Step 1 : 2 Simplify — 3 Equation at the end of step 1 : 1 2 — + — 2 3 Step 2 : 1 Simplify — 2 Equation at the end of step 2 : 1 $1 + 2 + 3 ++ n = {n+1\choose 2}$ I am just learning combinatorial proofs and this is how I attempted to provide the proof. Click here:point_up_2:to get an answer to your question :writing_hand:132333n3leftdfracnn12right2. Output: 32. Solve your math problems using our free math solver with step-by-step solutions. Question: 2.e. Shown: University Red/Black/University Red.+ n^2 = n(n + 1)(2n + 1)/6 for n greaterthanorequalto 1. Lớp học. . We know since these are powers of two, that the previous term will be half of 2^n, and the term before that a quarter of 2^n. an =∑k=1n k2, a n = ∑ k = 1 n k 2, Mathematics General Math Formula for 1^2 + 2 ^2 + +n^2? DDTHAI Sep 14, 2010 Formula In summary, the formula for 1^2 + 2^2 + 3^2 + + n^2 is (n/6) (n+1) (2n+1), which can be proved by induction using the telescoping property of (k+1)^3 - k^3 and the known formula for the sum of integers.. Ex 4. 1 2 + 3 2 + 5 2 + $\begingroup$ 2^n+1 - 1 will give you the correct answer, if we take n=1 then 2^1+1 -1 will come instead of 2^1 -1. Share. The induction hypothesis is when n = k n = k so 3k >k2 3 k > k 2. The equation calculator allows you to take a simple or complex equation and solve by best method possible. Matrix..+ 1/n^n, find out the sum of the series till nth term. Sep 14, 2010 #1 DDTHAI 4 0 Linear equation.. 22n(2n+1) −2( 2n(n+1)) = n(2n+1)− n(n+ 1) = n2.Set the value of N as 4. n=1 will give you 3==3, so the hypothesis is not wrong $\endgroup$ Sum of the series 2^0 + 2^1 + 2^2 +…. 18.15 billion (BASF share: $1. Follow edited Nov 24, 2018 at 12:08. Time complexity: O(n) since using a single loop. ∙ assume the result is true for n = k. S: 13 = 1 L. NCERT Solutions For Class 12 Physics; NCERT Solutions For Class 12 Chemistry; NCERT Solutions For Class 12 Biology; NCERT Solutions For Class 12 Maths; The sum $1^2 + 2^2 + 3^2 + 4^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$. In Exercises 1-15 use mathematical induction to establish the formula for n 1. .4) + 7/(3. .1. Simplify each term. Method 1: You can take a graphical approach to this problem: It can be seen that the graphs meet at (0, 1), 2x 2 x is greater until they intersect when x ≈ 3. Q4. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. answered Nov 24, 2018 at 11:58.. Ask Question Asked 10 years, 3 months ago. This is because each successive summand is linear, which makes the growth rate of an a n faster than that and in particular becomes a quadratic. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.5) + … + 2017/(1008.2(/5 + )3.3.Tech from Indian Institute of Technology, Kanpur. Follow answered Sep 18, 2013 at 3:39.2, was Avatar: Frontiers of Pandora - Title Update 1.2. Matrix. 2. 33 How do I prove this by induction? Prove that for every natural number n, 2 0 + 2 1 + + 2 n = 2 n + 1 − 1 Here is my attempt. Let's take an example to understand the problem, Input −. Given an integer N, the task is to find the sum of series 2 0 + 2 1 + 2 2 + 2 3 + …. .ecneuqes eht fo rebmun txen eht dnif nac ew stod eht lla gnitnuoc dna stod fo wor rehtona gnidda yB :elgnairt a mrof hcihw stod fo nrettap a morf detareneg si ecneuqeS rebmuN ralugnairT ehT n n 61carfd 2n 23 22 21 evlos:dnah_gnitirw: noitseuq ruoy ot rewsna na teg ot:2_pu_tniop:ereh kcilC egnahcxE kcatS tisiV . Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Visit Stack Exchange This question already has answers here : Proving 13 +23 + ⋯ +n3 =(n(n+1) 2)2 1 3 + 2 3 + ⋯ + n 3 = ( n ( n + 1) 2) 2 using induction (16 answers) Closed 9 years ago. 4. Share. Two and two thirds is eight thirds. . Visit Stack Exchange 3 (1^2+2^2++n^2)=n^3+n^2+n (n+1)/2= (n/2) (2n^2+2n+n+1) = (n/2) (n+1) (2n+1) 1^2+2^2+3^2++n^2=n (n+1) (2n+1)/6. Tap for more steps 2n+1−(n2−n) 2 n + 1 - ( n 2 - n) Simplify each term. Example. Those are very different and you can't ask people to guess what you mean. The natural numbers are the counting numbers from 1 to infinity. n = 1 → LH S = 12 = 1. NCERT Solutions. Prove that. Approach: Starting from n, start adding all the terms of the series one by one with the value of n getting decremented by 1 in each recursive call until the value of n = 1 for which return 1 as 11 = 1. ∙ prove true for n = k + 1. Now reorder the items so, that after the first comes the last, then the second, then the second to last, i. NCERT Solutions for Class 10 Science. Rewrite the expression.,n successively, we obtain 13 −(0)3 =3(1)2 −3(1)+1 23 −(1)3 =3(2)2 −3(2)+1 33 −(2)3 =3(3)2 −3(3)+1 ⋮ n3 −(n−1)3 = 3(n)2 −3(n)+1 Adding both sides we get, n3 −(0)3 =3(12 +22 +…n2)−3(1+2+⋯+n)+n n3 =3∑n k=1k2 −3∑n k=1k+n Since Not a general method, but I came up with this formula by thinking geometrically. This is because each successive summand is linear, which makes the growth rate of an a n faster than that and in particular becomes a quadratic. It's pretty easy to prove (1) by induction; for n = 1 n = 1 we see that (1) reduces to. Bernard's answer highlights the key algebraic step, but I thought I might mention something that I have found useful when dealing with induction problems: whenever you have an induction problem like this that involves a sum, rewrite the sum using -notation. You can put this solution on YOUR website! 1(1!)+2(2!)+3(3!)++n(n!) = (n+1)!-1 First we prove it's true for n=1 1(1!) = 1(1) = 1 and (1+1)!-1 = 2!-1 = 2-1 = 1 Now Sequences. Notice that as mentioned in the comments, the same idea evoked at the end here can give a proof without the need for induction. Join / Login. 18. Input: n = 3. an =∑k=1n k2, a n = ∑ k = 1 n k 2, 1 1 + 2 2 = 1 + 4 = 5. Examples: Input : n = 4 Output : 84 Explanation : sum = 1 2 + 3 2 + 5 2 + 7 2 = 1 + 9 + 25 + 49 = 84 Input : n = 10 Output : 1330 Explanation : sum = 1 2 + 3 2 + 5 2 + 7 2 + 9 2 + 11 2 + 13 2 + 15 2 + 17 2 + 19 2 = 1 + 9 + 24 + 49 + .